Orthogonal transformations

  1. daniel_i_l

    daniel_i_l 866
    Gold Member

    1. The problem statement, all variables and given/known data
    Given two sub-spaces of R^n - W_1 and W_2 where dimW_1 = dimW_2 =/= 0.
    Prove that there exists an orthogonal transformation T:R^n -> R^n so that
    T(W_1) = T(W_2)


    2. Relevant equations



    3. The attempt at a solution
    If dimW_1 = dimW_2 = m then we can say that {v_1,...,v_m} is a basis of W_1 and {u_1,...,u_m} is a basis of W_2. We can make from these bases of R^n:
    {v_1,...,v_m,a_1,...,a_{n-m}} is a basis of R^n and
    {u_1,...,u_m,b_1,...,b_{n-m}} is also.
    From these we can make orthonormal bases of R^n (via GS) so that:
    {v'_1,...,v'_m,a'_1,...,a'_{n-m}} is an orthonormal basis of R^n and
    {u'_1,...,u'_m,b'_1,...,b'_{n-m}} is also.
    Now we make a transformation where:
    T(v'_i) = u'_i
    T(a'_i) = b'_i
    Now, since the transformation of an orthonormal base gives another orthonormal base it's a ON transformation. And since {v'_1,...,v'_m} is a basis of W_1 and{u'_1,...,u'_m} is a basis of W_2 then T(W_1) = W_2
    Is that a complete proof?
    Thanks.
     
  2. jcsd
  3. Dick

    Dick 25,893
    Science Advisor
    Homework Helper

    That's pretty much it. Though you may want to point out that you've diligently applied GS in such a way as to ensure that (v'_1..v'_m) and (u'_1...u'_m) are still a basis for W_1 and W_2. I.e. what order to do GS in? You are probably already thinking of the correct order. I'm just saying this because I can't think of anything else to fault and am wondering why you bothered to post this in the first place. It is really easy.
     
    Last edited: Jun 29, 2007
  4. daniel_i_l

    daniel_i_l 866
    Gold Member

    Thanks for the comments :)
     
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